$\mathbb{Z}_q$-valued generalized bent functions in odd characteristics

TL;DR

This paper studies generalized bent functions from vector spaces over finite fields to cyclic groups, establishing conditions for their bent-ness and weak regularity, and providing related constructions.

Contribution

It introduces new necessary and sufficient conditions for generalized bent functions in odd characteristics, expanding understanding beyond classical Boolean functions.

Findings

Characterization of bent-ness in terms of classical p-ary bent functions

Sufficient conditions for weakly regular gbent functions when q is not a power of p

New constructions of generalized bent functions

Abstract

In this paper, we investigate properties of functions from $\mathbb{Z}_{p}^n$ to $\mathbb{Z}_q$, where $p$ is an odd prime and $q$ is a positive integer divided by $p$. we present the sufficient and necessary conditions for bent-ness of such generalized Boolean functions in terms of classical $p$-ary bent functions, when $q=p^k$. When $q$ is divided by $p$ but not a power of it, we give an sufficient condition for weakly regular gbent functions. Some related constructions are also obtained.